Three Dimensional Geometry
= 3 D Geometry = Axes There are three axes in 3-D Geometry , X , Y & Z . There are 8 quadrants in 3d axes . x-coordinate is the distance of the point from y-z plane . Division of a line lying in a plane of axes Suppose the XZ plane is dividing a line, it does so in the ratio -y1/y2 . Similarly for yz and xy planes ,too . Polar Spherical Co-ordinates Direction Cosines If α β γ are the angles made by a line with positive direction of X, Y and Z axes , then they are called as direction angles of the line . The cos of α , β & γ are called as direction cosines of the line . cos2α + cos2β + cos2γ = 1 ''' If l , m , n are direction cosines of a given line , and a,b,c are any real numbers such that :- l/a = m/b = n/c ; then a,b,c are called as direction ratios of a line . If P(x1,y1,z1) and Q(x2,y2,z2) then the direction Ratios of PQ are :- x2 - x1 , y2 - y1 , z2 - z1 If AB = ax + by + cz , then direction ratios of line AB are a, b ,c . Suppose Ray BD bisects angle ABC , then the direction ratios of Bisector BD are equal to sum of the Direction ratios of AB and BC The vector li + mj + nk is a unit vector always . while ai + bj + ck is not a unit vector . Angle between two Lines If a1 b1 c1 and a2 b2 c2 are the direction ratios of two given lines , then cos θ = a1.a2 / |a1||a2| or cos θ = l1l2 + mm2 + n1n2 Area of Triangle Area of triangle = 1/2 AB x AC = Lines = Vector Equation of a Line 1) If a line passes through a point and is parallel to another vector , then '''r = a + λb 2) In non-parametric form , rxa = rxb Cartesian Equation x - x1 / a = y - y1 / b = z - z1 / c {Derived from 2 point form } In 2 point form :- x - x1 / x1 - x2 = y - y1 / y1 - y2 = z - z1 / z1 - z2 a = x1 - x2 ; b = y1 - y2 ; c = z1 - z2 Note : For a point on the line , the Cartesian equation can be used to find the point coordinates . (equate the cartesian equation with a constant ) x = x1 + ka y = y2 + kb z = z2 + kc Where , k is a constant that decides the distance of a point from the point A , on the line . Collinearity of 3 points three points are collinear if a1/a2 = b1/b2 = c1/c2 Distance Formula Section and Mid-point Formula For Section formula you can either consider m : n or λ : 1 Skew Lines Shortest Distance between 2 lines in plane is given by :- - a1 b1 b2 / |b1 x b2| For intersecting lines , - a1 b1 b2 = 0 ... Shortest distance = 0 = Plane = Equation of a Plane r.n = a.n r.n = p For plane passing through intersection of two planes ; a + 'λ'b = 0 (Where a & b are the equations of the two planes) Angle between planes cosθ = n1.n2 / |n1||n2| Between line and plane : sinθ = n.b / |n||b| Co-planarity of two lines If angle between a line and plane is 0 , it lies in the plane . (This can be used to check coplanarity ) Distance between a Point & Plane l(AP) = (a.n - p) / n l(AP) = ( ax + by + cz + d ) / sqrt (a2 + b2 + c2) where (x,y,z) = coordinates of point in space . ax + by + cz + d is equation of plane . Tips and Tricks # In formulas dealing with ratios (e.g. cartesian equation of line , condition for parallel and perpendicular lines) direction ratios can be replaced with direction cosines . Category:Mathematics